Heating long pipes

In this paper, we study the steady-state distribution of heat on long pipes in R3 heated along some regions of their surfaces. In particular, we prove that, if the pipe P={(x,y,z): x2+y2<1} is heated along its surface belt B(a)={(x,y,z): x2+y2=1,-a<z<a}, a>0, then the temperature in its cross-sections Dc={(x,y,z)is an element of P: z=c} is increasing in the radial direction for all c in the interval [-a,a].